## Thermodynamics and economics

If you’ve ever studied thermodynamics, you too were probably struck by the sheer abstractness and generality of its laws/postulates. It manages to build a rich physical theory (or perhaps a “framework” is a better description) based merely on assuming the existence of a handful of functions and quantities with some very general properties. The natural question seems to be, is it possible to give those axioms an interpretation other than the intended one (where, for example, $T$ roughly corresponds to our pre-theoretical notion of temperature)? If it is, one would (or at least I would) expect the alternative interpretation to deal with phenomena which are still statistical in nature (we assume that the fluctuations in different parts of the system are averaged out, spatiotemporally in thermodynamics), unidirectional and governed by a general class of quantitative relationships rather than one specific equation.

Well, I don’t know the first thing about economics, but it was the first thing to come to my mind. And it turns out my hunch was correct, there is indeed a portion of economics which can be axiomatized in essentially the same way as thermodynamics, as shown by E. Smith and D.K. Foley in their paper Classical thermodynamics and economic general equilibrium theory. (It may be worthwhile to think about how you would draw an analogy between the two before continuing.)

The role of thermodynamic systems in the formalism is taken over by the so-called quasi-linear neoclassical economies, which means we assume that each agent $j$ is assigned a vector of n+1 commodities $x$ (a commodity bundle) and tries to maximize his (quasiconcave, differentiable) ordinal utility function $u^j$ by exchanging goods with others. Quasi-linearity requires that each agent’s utility depend on one of the commodities $x^j_0$ (called the “linear commodity” or “money”) as: $u^j(x^j) = x^j_0 + \overline{u}^j(\overline{x}^j)$ (where $x^j = (x^j_0,\overline{x}^j)$). We will only be concerned with economic exchange, which corresponds to assuming the conservation of commodities (energy1). If any two such (groups of) agents are brought into “thermal contact” (are allowed to exchange their commodities), they will ideally keep trading their commodities until they reach a state of equilibrium where no two agents can both increase their utility by trading their commodities – a Pareto optimal state. The final Pareto optimal state given some initial endowment of goods is in general non-unique, but as it turns out, in a quasi-linear economy, the allocation of all non-linear commodities is the same for all such Pareto optimal allocations. In such an economic equilibrium, there is an economy-wide price for every commodity (otherwise, one could get richer by buying cheap and selling dear). And this price is uniquely defined by the distribution of commodities (it’s the common value of the agents’ marginal utilities at that distribution).

In fact, in a quasi-linear economy, we can aggregate the agents’ individual endowments and postulate that the equilibrium state maximizes their total utility from the non-linear goods, and the price $p_i$ of commodity $i$ is the intensive variable conjugate to the (obviously extensive) total amount of the commodity $x_i$ in the economy. The total utility is equivalent to entropy, the total amount of commodity $i$ is equivalent to the extensive variable $X$ (say, internal energy), and the price of commodity $i$ is equivalent to the (entropic) conjugate intensive variable to $X$ (say, the inverse of the temperature). An economic reinterpretation can also be found for the Legendre transforms of internal energy (see the paper).

1 The authors draw an analogy between $x_i$ and energy/volume, but given that there are $n$ different commodities, an analogy between $x_i$ and the number of moles $n_i$ seems more natural to my (lay) eye, given the absence of any a priori difference between the non-linear commodities and the obvious analogy between the nature of the two quantities. It would also mean we don’t have to assume that the utility function is increasing in the commodity or worry about the third law of thermodynamics (I don’t see any meaningful way to interpret it economically and the authors don’t even mention it).

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